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• Truth

Index-listp

Signature

(index-listp x numvars) → *

Arguments

numvars — Guard (natp numvars).

Definitions and Theorems

Function: index-listp

(defun index-listp (x numvars)
       (declare (xargs :guard (natp numvars)))
       (let ((__function__ 'index-listp))
            (declare (ignorable __function__))
            (if (atom x)
                (eq x nil)
                (and (natp (car x))
                     (< (car x) (lnfix numvars))
                     (index-listp (cdr x) numvars)))))

Theorem: natp-nth-of-index-listp

(defthm natp-nth-of-index-listp
        (implies (and (index-listp x numvars)
                      (< (nfix n) (len x)))
                 (natp (nth n x))))

Theorem: nfix-nth-in-index-list-bound

(defthm nfix-nth-in-index-list-bound
        (implies (and (index-listp x numvars)
                      (posp numvars))
                 (< (nfix (nth n x)) numvars))
        :rule-classes (:rewrite :linear))

Theorem: nth-in-index-list-bound

(defthm nth-in-index-list-bound
        (implies (and (index-listp x numvars)
                      (natp numvars)
                      (natp (nth n x)))
                 (< (nth n x) numvars))
        :rule-classes (:rewrite :linear))

Theorem: nat-listp-when-index-listp

(defthm nat-listp-when-index-listp
        (implies (index-listp x numvars)
                 (nat-listp x)))

Theorem: true-listp-when-index-listp

(defthm true-listp-when-index-listp
        (implies (index-listp x numvars)
                 (true-listp x))
        :rule-classes :forward-chaining)

Theorem: index-listp-of-nfix-numvars

(defthm index-listp-of-nfix-numvars
        (equal (index-listp x (nfix numvars))
               (index-listp x numvars)))

Theorem: index-listp-nat-equiv-congruence-on-numvars

(defthm index-listp-nat-equiv-congruence-on-numvars
        (implies (nat-equiv numvars numvars-equiv)
                 (equal (index-listp x numvars)
                        (index-listp x numvars-equiv)))
        :rule-classes :congruence)